The Wiener-Khinchin theorem in temporal spectral estimation states that the power spectral density (PSD) of a wide sense stationary (WSS) signal and its autocorrelation function (ACF), \(r_{xx}[k]\) , are related by the Fourier transform [1]}. The PPO makes the Fourier transform pair with a spatial ACF estimate, \(\hat{r}[k]\) . With \(k=m-n\) , the expression for \(\hat{P}(u)\)  in (REF ) can be rewritten as \(\hat{P}(u)\)
\(\begin{split}= &\sum _{k=-(N_e-1)}^{M_e-1}\sum _{l}(w_1[l]x_1[l])(w_2[l-k]x_2[l-k])^*\dfrac{e^{-j\pi uk}}{\nu } \\= &\sum _{k=-(N_e-1)}^{M_e-1}\dfrac{(w_1[k]x_1[k])\star (w_2[-k]x_2[-k])^*}{\nu }e^{-j\pi uk},\end{split}\) 
